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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Summability in Topological Spaces
H. Çakalli, M. Kazim KhanMaltepe Univ., Istanbul, Turkey, Kent
State Univ., Kent Ohio.
University of North Florida, Jacksonville, FL. October 29-30,
2010
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Outline
1 Summability Methods
2 The Setup
3 A Bit of History
4 Abelian Side
5 Tauberian Side
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Applications of Summability Methods
Summability theory has historically been concerned with the
notion ofassigning a limit to a linear space-valued sequences,
especially if thesequence is divergent.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Applications of Summability Methods
Summability theory has historically been concerned with the
notion ofassigning a limit to a linear space-valued sequences,
especially if thesequence is divergent.
This underlying linearity naturally leads to the use of matrices
as potentialsummability methods. However, in general, a summability
method need notbe a matrix method.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Applications of Summability Methods
Summability theory has historically been concerned with the
notion ofassigning a limit to a linear space-valued sequences,
especially if thesequence is divergent.
This underlying linearity naturally leads to the use of matrices
as potentialsummability methods. However, in general, a summability
method need notbe a matrix method.
Most of the famous applications therefore are cast in exactly
this context. Forinstance,
The weak and the strong laws of large numbers of probability
theory.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Applications of Summability Methods
Summability theory has historically been concerned with the
notion ofassigning a limit to a linear space-valued sequences,
especially if thesequence is divergent.
This underlying linearity naturally leads to the use of matrices
as potentialsummability methods. However, in general, a summability
method need notbe a matrix method.
Most of the famous applications therefore are cast in exactly
this context. Forinstance,
The weak and the strong laws of large numbers of probability
theory.
Fejer’s theorem on convergence of Fourier series.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Applications of Summability Methods
Summability theory has historically been concerned with the
notion ofassigning a limit to a linear space-valued sequences,
especially if thesequence is divergent.
This underlying linearity naturally leads to the use of matrices
as potentialsummability methods. However, in general, a summability
method need notbe a matrix method.
Most of the famous applications therefore are cast in exactly
this context. Forinstance,
The weak and the strong laws of large numbers of probability
theory.
Fejer’s theorem on convergence of Fourier series.
Komlos’ theorem for L1-bounded sequences
And so on · · · .
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Outline Summability Methods The Setup A Bit of History Abelian
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Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
Answer: Out of the four classical summability methods, only one
of themrequires neither the “addition” operation nor the “partial
order” concept. In thissense it is the most primitive of them
all.
More precisely, consider the following classical summability
methods.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
Answer: Out of the four classical summability methods, only one
of themrequires neither the “addition” operation nor the “partial
order” concept. In thissense it is the most primitive of them
all.
More precisely, consider the following classical summability
methods.
(a) Strong convergence,
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
Answer: Out of the four classical summability methods, only one
of themrequires neither the “addition” operation nor the “partial
order” concept. In thissense it is the most primitive of them
all.
More precisely, consider the following classical summability
methods.
(a) Strong convergence,
(b) Statistical convergence,
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
Answer: Out of the four classical summability methods, only one
of themrequires neither the “addition” operation nor the “partial
order” concept. In thissense it is the most primitive of them
all.
More precisely, consider the following classical summability
methods.
(a) Strong convergence,
(b) Statistical convergence,
(c) Distributional convergence,
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Four Summability Methods
Problem (Four Types of Summability Methods)
The question is: “How do you introduce summability notion in a
generaltopological spaces where there is no binary operation of
“addition” nor anynatural partial order?
Answer: Out of the four classical summability methods, only one
of themrequires neither the “addition” operation nor the “partial
order” concept. In thissense it is the most primitive of them
all.
More precisely, consider the following classical summability
methods.
(a) Strong convergence,
(b) Statistical convergence,
(c) Distributional convergence,
(d) classical matrix summability.
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Outline Summability Methods The Setup A Bit of History Abelian
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A-Strong and A-Stat Convergence
Thoughout assume that A = [ank ] is a nonnegative regular
summabilitymethod. Not much loss takes place to assume that the row
sums equal toone.
Definition (A-strong convergence)
We say that x = (xk ) is A-strongly summable to α if
limn→∞
X
k
|xk − α| ank = 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Strong and A-Stat Convergence
Thoughout assume that A = [ank ] is a nonnegative regular
summabilitymethod. Not much loss takes place to assume that the row
sums equal toone.
Definition (A-strong convergence)
We say that x = (xk ) is A-strongly summable to α if
limn→∞
X
k
|xk − α| ank = 0.
Definition (A-stat convergence)
We say x = (xk ) is A-statistically convergent to α if for any ǫ
> 0, we have
limn→∞
X
k :|xk−α|≥ǫ
ank = 0.
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Outline Summability Methods The Setup A Bit of History Abelian
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A-Dist Convergence & A-Suammbility
Definition (A-distributional convergence)
If x is a real sequence, we say x is A-distributionaly
convergent to F , where Fis a probability distribution on ℜ and
limn→∞
X
k :xk≤t
ank = F (t),
for all t in the continuity set of the distribution F .
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Outline Summability Methods The Setup A Bit of History Abelian
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A-Dist Convergence & A-Suammbility
Definition (A-distributional convergence)
If x is a real sequence, we say x is A-distributionaly
convergent to F , where Fis a probability distribution on ℜ and
limn→∞
X
k :xk≤t
ank = F (t),
for all t in the continuity set of the distribution F .
Definition (A-summability)
Finally, we say that x is A summable to α if
limn→∞
X
k
xk ank = α.
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Outline Summability Methods The Setup A Bit of History Abelian
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A-statistical convergence
At a first glance all of the four methods seem to be using a
linear, or group ororder structure. For instance, the classic
matrix summability uses group
structure having an operation of addition. The A-distributional
convergenceuses order.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-statistical convergence
At a first glance all of the four methods seem to be using a
linear, or group ororder structure. For instance, the classic
matrix summability uses group
structure having an operation of addition. The A-distributional
convergenceuses order.
The remaining two, A-strong convergence and A-stat convergence,
usedistance structure since they both use
‖xk − α‖, ρ(xk , α).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-statistical convergence
At a first glance all of the four methods seem to be using a
linear, or group ororder structure. For instance, the classic
matrix summability uses group
structure having an operation of addition. The A-distributional
convergenceuses order.
The remaining two, A-strong convergence and A-stat convergence,
usedistance structure since they both use
‖xk − α‖, ρ(xk , α).
This then leads one to consider general topological structures
by replacingρ(xk , α) ≥ ǫ by its natural counterpart,
xk 6∈ Uα,
where Uα is any open set containing α. So, how do you bring the
summability
structure into the topological space?
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Outline Summability Methods The Setup A Bit of History Abelian
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Mathematical Structure
Let (X ,B, τ ) be any topological space, where B is the Borel
sigma fieldgenerated by the open sets. In order to define a
summability notion in X , wewill inject several probability
measures µn defined over B with the help of anonnegative regular
summability matrix A = (ank).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Mathematical Structure
Let (X ,B, τ ) be any topological space, where B is the Borel
sigma fieldgenerated by the open sets. In order to define a
summability notion in X , wewill inject several probability
measures µn defined over B with the help of anonnegative regular
summability matrix A = (ank).Consider ([0, 1],M, λ) be the usual
Lebesgue measure. Partition the interval[0, 1] by An,0 = [0, an0),
and
An,k =
2
4
k−1X
j=0
anj ,k
X
j=0
anj
1
A , k = 1, 2, · · · .
Let fn : [0, 1] → N := {0, 1, 2 · · · }, where fn(ω) = k for ω ∈
An,k . Over thesigma field of powerset of N this fn induced a
measure νn defined byνn(k) = λ(An,k) = ank .
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Mathematical Structure
Let (X ,B, τ ) be any topological space, where B is the Borel
sigma fieldgenerated by the open sets. In order to define a
summability notion in X , wewill inject several probability
measures µn defined over B with the help of anonnegative regular
summability matrix A = (ank).Consider ([0, 1],M, λ) be the usual
Lebesgue measure. Partition the interval[0, 1] by An,0 = [0, an0),
and
An,k =
2
4
k−1X
j=0
anj ,k
X
j=0
anj
1
A , k = 1, 2, · · · .
Let fn : [0, 1] → N := {0, 1, 2 · · · }, where fn(ω) = k for ω ∈
An,k . Over thesigma field of powerset of N this fn induced a
measure νn defined byνn(k) = λ(An,k) = ank .Any function x : N → X
is automatically 2N/B measureable. Now considerthe sequence of
compositions
x(fn) : [0, 1] → X , with x(k) = xk ∈ X .
This brings with it a sequence of measures µn over B. Note
that
µn(B) := λ(x(fn) ∈ B) = λ(fn ∈ x−1(B)) =X
j∈x−1(B)
anj =X
j:xj∈B
anj .
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Mathematical Structure
In fact, in the last setup we may as well consider double
arrays, if we like,without encountering much difficulties. That is,
let x (n) : N → X withx (n)(k) = xnk ∈ X .
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Mathematical Structure
In fact, in the last setup we may as well consider double
arrays, if we like,without encountering much difficulties. That is,
let x (n) : N → X withx (n)(k) = xnk ∈ X .
-
?
QQ
QQ
QQ
Qs
([0, 1],M, λ) (N, 2N, νn)
(X ,B, µn)
fn
x (n)x (n)(fn)
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Mathematical Structure
In fact, in the last setup we may as well consider double
arrays, if we like,without encountering much difficulties. That is,
let x (n) : N → X withx (n)(k) = xnk ∈ X .
-
?
QQ
QQ
QQ
Qs
([0, 1],M, λ) (N, 2N, νn)
(X ,B, µn)
fn
x (n)x (n)(fn)
So, we get
µn(B) := λ(x(n)(fn) ∈ B) =
X
j:xnj∈B
anj , for all B ∈ B.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical Convergence
So for any nonnegative regular summability matrix A = [ank ]
with row sumsone, and any double array x = (xnk ) in X , we get a
sequence of measurespaces
(X ,B, µn), µn(B) =X
j:xnj∈B
anj , B ∈ B.
We say x = (xnk ) is A-statistically convergent to α ∈ X if for
every open setUα that contains α, we have
limn→∞
µn(Ucα) = 0.
To avoid the usual non-uniqueness issues, we will assume
throughout thatthe space X is at least T2 (Hausforff).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical Convergence
So for any nonnegative regular summability matrix A = [ank ]
with row sumsone, and any double array x = (xnk ) in X , we get a
sequence of measurespaces
(X ,B, µn), µn(B) =X
j:xnj∈B
anj , B ∈ B.
We say x = (xnk ) is A-statistically convergent to α ∈ X if for
every open setUα that contains α, we have
limn→∞
µn(Ucα) = 0.
To avoid the usual non-uniqueness issues, we will assume
throughout thatthe space X is at least T2 (Hausforff).By the way,
one could introduce ideals, instead of the A-density zero sets.Our
goal here is to see separation of notions injected by summability
methodA versus inherent topological notions.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical Convergence
So for any nonnegative regular summability matrix A = [ank ]
with row sumsone, and any double array x = (xnk ) in X , we get a
sequence of measurespaces
(X ,B, µn), µn(B) =X
j:xnj∈B
anj , B ∈ B.
We say x = (xnk ) is A-statistically convergent to α ∈ X if for
every open setUα that contains α, we have
limn→∞
µn(Ucα) = 0.
To avoid the usual non-uniqueness issues, we will assume
throughout thatthe space X is at least T2 (Hausforff).By the way,
one could introduce ideals, instead of the A-density zero sets.Our
goal here is to see separation of notions injected by summability
methodA versus inherent topological notions.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
In todays talk we will address the following
questions/issues.
(i) Is this notion regular?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
In todays talk we will address the following
questions/issues.
(i) Is this notion regular?
(ii) As Fridy showed, for real/complex sequences statistical
convergencecan be characterized through a convergent subsequence
outside a setof density zero. Does such a characterization hold for
arbitrary T2topological spaces?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
In todays talk we will address the following
questions/issues.
(i) Is this notion regular?
(ii) As Fridy showed, for real/complex sequences statistical
convergencecan be characterized through a convergent subsequence
outside a setof density zero. Does such a characterization hold for
arbitrary T2topological spaces?
(iii) What kind of Abelian theory does this spawn?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
In todays talk we will address the following
questions/issues.
(i) Is this notion regular?
(ii) As Fridy showed, for real/complex sequences statistical
convergencecan be characterized through a convergent subsequence
outside a setof density zero. Does such a characterization hold for
arbitrary T2topological spaces?
(iii) What kind of Abelian theory does this spawn?
(iv) And of course, what kind of Tauberian theory does this
spawn?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: Summability in topological groups
During 1967-1969 Prullage wrote a series of articles involving
ordinarysummability notions in topological groups.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: Summability in topological groups
During 1967-1969 Prullage wrote a series of articles involving
ordinarysummability notions in topological groups.
Around 1995, 1996 Çakalli studied lacunary statistical
convergence intopological groups where its convergence field is
compared with theconvergence field of Cesàro statistical
convergence.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: Summability in topological groups
During 1967-1969 Prullage wrote a series of articles involving
ordinarysummability notions in topological groups.
Around 1995, 1996 Çakalli studied lacunary statistical
convergence intopological groups where its convergence field is
compared with theconvergence field of Cesàro statistical
convergence.
Around 2005 Lahiri and Das took this notion in the language
ofI-convergence, which is a bit more generalized form of
statisticalconvergence, in which the ideal, I, may not satisfy all
the properties of theclass of sets of A-density zero.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: Summability in topological groups
During 1967-1969 Prullage wrote a series of articles involving
ordinarysummability notions in topological groups.
Around 1995, 1996 Çakalli studied lacunary statistical
convergence intopological groups where its convergence field is
compared with theconvergence field of Cesàro statistical
convergence.
Around 2005 Lahiri and Das took this notion in the language
ofI-convergence, which is a bit more generalized form of
statisticalconvergence, in which the ideal, I, may not satisfy all
the properties of theclass of sets of A-density zero.
All of the above references are concerned with the nature of the
convergencefield. Of course summability theory goes in two opposite
directions — theAbelian side and the Tauberian side —.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Abelian side
Statistical convergence, contrary to usual attributions, seems
to have beenused by R. C. Buck in 1946, although he did not give it
the current name of“statistical convegence”.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Abelian side
Statistical convergence, contrary to usual attributions, seems
to have beenused by R. C. Buck in 1946, although he did not give it
the current name of“statistical convegence”.
Actually probabilists would object to this attribution on the
grounds thatstatistical convergence is a very special notion of
convergence in probability(to a constant) and hence was in the
literature for atleast fifty years prior toBuck when Chebyshev
proved the Weak Law of Large Numbers. Anways, in1951 H. Fast gave
it the name of “statistical convegence” and thenSchoenberg, Salat
and others picked up and popularized this name.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Abelian side
Statistical convergence, contrary to usual attributions, seems
to have beenused by R. C. Buck in 1946, although he did not give it
the current name of“statistical convegence”.
Actually probabilists would object to this attribution on the
grounds thatstatistical convergence is a very special notion of
convergence in probability(to a constant) and hence was in the
literature for atleast fifty years prior toBuck when Chebyshev
proved the Weak Law of Large Numbers. Anways, in1951 H. Fast gave
it the name of “statistical convegence” and thenSchoenberg, Salat
and others picked up and popularized this name.
Towards the Abelian direction, A-statistical convergence raises
thefundamental issue of whether it can be characterized via a
convergentsubsequence whose indicies form a set of A-density one.
It is not difficult toshow that over metric spaces this is possible
along the same lines as shownby Fridy. We will have a bit more to
say for topological spaces here.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
When only a metric structure is available the two-sided
Tauberiantheorems exist, at least for statistical convergence.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
When only a metric structure is available the two-sided
Tauberiantheorems exist, at least for statistical convergence.
Then there are gap Tauberian theorems that seem to exist in
parallel tothe above two varieties. However, what seems to be
missed is that, forstatistical convergence, neither the linear
structure nor the metricstructure are at its core.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
When only a metric structure is available the two-sided
Tauberiantheorems exist, at least for statistical convergence.
Then there are gap Tauberian theorems that seem to exist in
parallel tothe above two varieties. However, what seems to be
missed is that, forstatistical convergence, neither the linear
structure nor the metricstructure are at its core.
A classic result of Paul Erdös says that gap Tauberian theorems
need notexist for matrix methods. The classic example being the
Borel method. Whengap Tauberian theorems do exist, the Tauberian
condition is intimatelydependent on the underlying row structure of
the summability matrix.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
When only a metric structure is available the two-sided
Tauberiantheorems exist, at least for statistical convergence.
Then there are gap Tauberian theorems that seem to exist in
parallel tothe above two varieties. However, what seems to be
missed is that, forstatistical convergence, neither the linear
structure nor the metricstructure are at its core.
A classic result of Paul Erdös says that gap Tauberian theorems
need notexist for matrix methods. The classic example being the
Borel method. Whengap Tauberian theorems do exist, the Tauberian
condition is intimatelydependent on the underlying row structure of
the summability matrix.Statistical Tauberian theory, although is
distinctly different fromlinear/matrix-Tauberian theory, the two
share some common features.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
History: The Tauberian side
Tauberian theory has at least three major varieties.
Over spaces with linear and order structure the one-sided
Tauberiantheory has been studied for over one hundred years and the
theory ismost refined.
When only a metric structure is available the two-sided
Tauberiantheorems exist, at least for statistical convergence.
Then there are gap Tauberian theorems that seem to exist in
parallel tothe above two varieties. However, what seems to be
missed is that, forstatistical convergence, neither the linear
structure nor the metricstructure are at its core.
A classic result of Paul Erdös says that gap Tauberian theorems
need notexist for matrix methods. The classic example being the
Borel method. Whengap Tauberian theorems do exist, the Tauberian
condition is intimatelydependent on the underlying row structure of
the summability matrix.Statistical Tauberian theory, although is
distinctly different fromlinear/matrix-Tauberian theory, the two
share some common features.Over topological spaces, as we will see,
gap Tauberian theory happens to bethe most natural thing to
build.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
Why is A-stat convergence regular in a topological space?
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Outline Summability Methods The Setup A Bit of History Abelian
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A-Statistical convergence
Why is A-stat convergence regular in a topological space?The
answer is easy. Yes. If xk is convergent to α in X , then for any
open setUα containing α, we can find an N such that xk ∈ Uα for all
k > N. Therefore,
µn(Ucα) =
X
k :xk 6∈Ucα
ank ≤N
X
k=0
ank .
Since A = [ank ] is regular, we see that
limn→∞
µn(Ucα) ≤ lim
n→∞
NX
k=0
ank = 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
A-Statistical convergence
Why is A-stat convergence regular in a topological space?The
answer is easy. Yes. If xk is convergent to α in X , then for any
open setUα containing α, we can find an N such that xk ∈ Uα for all
k > N. Therefore,
µn(Ucα) =
X
k :xk 6∈Ucα
ank ≤N
X
k=0
ank .
Since A = [ank ] is regular, we see that
limn→∞
µn(Ucα) ≤ lim
n→∞
NX
k=0
ank = 0.
When X happens to be metrizable, with metric ρ, this notion can
be written asfollows. For any ǫ > 0 there exists an N so
that
limn→∞
X
k : ρ(xk ,α)>ǫ
ank = 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Density convergence property
Let A be a nonnegative regular summability method and let x =
(xk ) be asequence taking values in a T2 topological space X . If
there exists a setE ⊆ N such that
δA(E) := limn→∞
X
k∈E
ank = 0,
and x is convergent to some α over Ec , then we will say that x
has A-densityconvergence property (DCP(A) for short).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Density convergence property
Let A be a nonnegative regular summability method and let x =
(xk ) be asequence taking values in a T2 topological space X . If
there exists a setE ⊆ N such that
δA(E) := limn→∞
X
k∈E
ank = 0,
and x is convergent to some α over Ec , then we will say that x
has A-densityconvergence property (DCP(A) for short).
It is easy to see that if x has the DCP(A) then x is
A-statistically convergent toα, where α is its subsequential limit
over its Ec .
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Density convergence property
Let A be a nonnegative regular summability method and let x =
(xk ) be asequence taking values in a T2 topological space X . If
there exists a setE ⊆ N such that
δA(E) := limn→∞
X
k∈E
ank = 0,
and x is convergent to some α over Ec , then we will say that x
has A-densityconvergence property (DCP(A) for short).
It is easy to see that if x has the DCP(A) then x is
A-statistically convergent toα, where α is its subsequential limit
over its Ec .
The issue is whether the converse can hold. This is partially
addressed bythe following theorem.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Density convergence property
Theorem (DCP(A) vs. A-stat convergence)
Let X be a topological space and let α ∈ X have a countable
base. Then forany nonnegative regular summability matrix A, any
A-statistically convergentsequence to α has the DCP(A).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Density convergence property
Theorem (DCP(A) vs. A-stat convergence)
Let X be a topological space and let α ∈ X have a countable
base. Then forany nonnegative regular summability matrix A, any
A-statistically convergentsequence to α has the DCP(A).
We are unable to drop the assumption on the countability of the
base of α,however one can construct examples outside the
countability condition. Thegeneral problem seems to be still open
over arbitrary T2 spaces andnonnegative regular matrices A.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
DCP is a topological property
The next result shows that the DCP is a topological
property.
Theorem
Let X , Y be homeomorphic topological spaces, and let A be any
nonnegativeregular matrix. If every A-statistically convergent
sequence in X has theDCP(A) then every A-statistically convergent
sequence in Y also has theDCP(A).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap Tauberian condition
Let γ : N → N denote an increasing function with γ(0) = 0.
Let
G(γ) = {x = (xk ) : xk 6= xk+1implies there exists r ∈ N such
that k = γ(r)}
Definition
For a nonnegative regular matrix A, we say that G(γ) is an
A-statistical gapTauberian condition if x ∈ G(γ) and x is
A-statistically convergent to some αtogether imply that x is
convergent.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
Our first result in the Tauberian direction shows topological
invariance forstatistical gap Tauberian theorems. That is they do
not depend on theunderlying topological structure at all. They are
truely controlled by thesummability method used!
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
Our first result in the Tauberian direction shows topological
invariance forstatistical gap Tauberian theorems. That is they do
not depend on theunderlying topological structure at all. They are
truely controlled by thesummability method used!
Theorem
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for real
valuedsequences.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
Our first result in the Tauberian direction shows topological
invariance forstatistical gap Tauberian theorems. That is they do
not depend on theunderlying topological structure at all. They are
truely controlled by thesummability method used!
Theorem
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for real
valuedsequences.
G(γ) is an A-statistical gap Tauberian condition for any
Hausdorfftopological space valued sequences.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
An idea of Connor (1993) can now be used to get the
followingcharacterizations. For a metric space valued sequence x =
(xk ) we say x isstrongly A-summable to α if
limn→∞
∞X
k=0
ankρ(xk , α) = 0.
Corollary
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for any T2
topologicalspace valued sequences.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
An idea of Connor (1993) can now be used to get the
followingcharacterizations. For a metric space valued sequence x =
(xk ) we say x isstrongly A-summable to α if
limn→∞
∞X
k=0
ankρ(xk , α) = 0.
Corollary
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for any T2
topologicalspace valued sequences.
G(γ) is a gap Tauberian condition for A-strong convergence for
metricspaces.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
An idea of Connor (1993) can now be used to get the
followingcharacterizations. For a metric space valued sequence x =
(xk ) we say x isstrongly A-summable to α if
limn→∞
∞X
k=0
ankρ(xk , α) = 0.
Corollary
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for any T2
topologicalspace valued sequences.
G(γ) is a gap Tauberian condition for A-strong convergence for
metricspaces.
For all increasing subsequences of {nr} of natural numbers,
lim supn
X
r
X
k∈(γ(nr ),γ(nr +1)]
ank > 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Topological invariance
An idea of Connor (1993) can now be used to get the
followingcharacterizations. For a metric space valued sequence x =
(xk ) we say x isstrongly A-summable to α if
limn→∞
∞X
k=0
ankρ(xk , α) = 0.
Corollary
Let A be a nonnegative regular matrix. The following statements
areequivalent.
G(γ) is an A-statistical gap Tauberian condition for any T2
topologicalspace valued sequences.
G(γ) is a gap Tauberian condition for A-strong convergence for
metricspaces.
For all increasing subsequences of {nr} of natural numbers,
lim supn
X
r
X
k∈(γ(nr ),γ(nr +1)]
ank > 0.
So the race is on: find these γ(k) for various classical
summability methods.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions
In 1993 Connor had already provided such a function, γ, for the
regular Rieszmethods.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions
In 1993 Connor had already provided such a function, γ, for the
regular Rieszmethods.In the next three results provide the
appropriate gap functions of theTauberian theorems for most of the
classical summability methods, such asthe Euler-Borel class and the
Hausdorff class. The following is an extensionof Fridy’s gap
Tauberian theorem.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions
In 1993 Connor had already provided such a function, γ, for the
regular Rieszmethods.In the next three results provide the
appropriate gap functions of theTauberian theorems for most of the
classical summability methods, such asthe Euler-Borel class and the
Hausdorff class. The following is an extensionof Fridy’s gap
Tauberian theorem.
Theorem
Let {k(1), k(2), · · · } be an increasing sequence of positive
integers such that
lim infi
k(i + 1)k(i)
> 1, (1)
and let x be a sequence in a topological space such that x
remains constantover the gaps (k(i), k(i + 1)]. If x is
C1-statistical convergent to α then xconverges to α.
Here C1 stands for the Cesàro matrix.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions
In 1993 Connor had already provided such a function, γ, for the
regular Rieszmethods.In the next three results provide the
appropriate gap functions of theTauberian theorems for most of the
classical summability methods, such asthe Euler-Borel class and the
Hausdorff class. The following is an extensionof Fridy’s gap
Tauberian theorem.
Theorem
Let {k(1), k(2), · · · } be an increasing sequence of positive
integers such that
lim infi
k(i + 1)k(i)
> 1, (1)
and let x be a sequence in a topological space such that x
remains constantover the gaps (k(i), k(i + 1)]. If x is
C1-statistical convergent to α then xconverges to α.
Here C1 stands for the Cesàro matrix. In fact, as the following
theoremshows, the Cesàro matrix can be replaced by a general
nonnegative regularHausdorff matrix.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions: Hausdorff
Theorem
Let Hφ be a regular Hausdorff method with a nondecreasing weight
functionφ. Again assume
lim infi
k(i + 1)k(i)
> 1,
holds. If x is a sequence in a topological space such that x
remains constantover the gaps (k(i), k(i + 1)] and if x is
Hφ-statistical convergent to α then xconverges to α.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions: Hausdorff
Theorem
Let Hφ be a regular Hausdorff method with a nondecreasing weight
functionφ. Again assume
lim infi
k(i + 1)k(i)
> 1,
holds. If x is a sequence in a topological space such that x
remains constantover the gaps (k(i), k(i + 1)] and if x is
Hφ-statistical convergent to α then xconverges to α.
In this theorem we may take γ(t) = ct for any constant c > 1.
The Tauberiancondition can be improved if the weight function φ of
the Hausdorff methodhas a point of jump.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions: Hausdorff with jumps
Theorem
Let Hφ be a regular Hausdorff method with a nondecreasing weight
functionφ, having a point of jump at some r ∈ (0, 1). Let {k(1),
k(2), · · · } be anincreasing sequence of positive integers such
that
lim infi
k(i + 1) − k(i)p
k(i)> 0. (2)
If x is a sequence in a topological space such that x remains
constant overthe gaps (k(i), k(i + 1)] and if x is Hφ-statistically
convergent to α then xconverges to α.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions: Hausdorff with jumps
Theorem
Let Hφ be a regular Hausdorff method with a nondecreasing weight
functionφ, having a point of jump at some r ∈ (0, 1). Let {k(1),
k(2), · · · } be anincreasing sequence of positive integers such
that
lim infi
k(i + 1) − k(i)p
k(i)> 0. (2)
If x is a sequence in a topological space such that x remains
constant overthe gaps (k(i), k(i + 1)] and if x is Hφ-statistically
convergent to α then xconverges to α.
In this result we may take γ(t) = ct2 with c > 0. This
theorem, in particular,provides a Tauberian theorem for the
Euler-statistical convergence. Since theEuler method is also a
member of the convolution methods, it is natural tosuspect that it
may have an analog for the convolution methods. This isindeed the
case.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Gap conditions: Convolution methods
Theorem
Let {k(1), k(2), · · · } be an increasing sequence of positive
integers satisfying(2), and let A = [ank ] be a regular convolution
method with finite variance. If xis a sequence in a topological
space such that x remains constant over thegaps (k(i), k(i + 1)]
and if x is A-statistically convergent to α then xconverges to
α.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
In the end we point out a somewhat interesting phenomenon
regarding thegap Tauberian rates and the gaps in the lacunary
statistical convergence.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
In the end we point out a somewhat interesting phenomenon
regarding thegap Tauberian rates and the gaps in the lacunary
statistical convergence.
Recall that a sequence θ = (kr ) of positive integers, such that
k0 = 0 andkr − kr−1 → ∞, is called a lacunary sequence.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
In the end we point out a somewhat interesting phenomenon
regarding thegap Tauberian rates and the gaps in the lacunary
statistical convergence.
Recall that a sequence θ = (kr ) of positive integers, such that
k0 = 0 andkr − kr−1 → ∞, is called a lacunary sequence.
We say that x = (xk ) is lacunary statistically convergent to α
if for each openset U containing α, we have
limr→∞
|{k ∈ (kr−1, kr ] : xk 6∈ U}|kr − kr−1
= 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
In the end we point out a somewhat interesting phenomenon
regarding thegap Tauberian rates and the gaps in the lacunary
statistical convergence.
Recall that a sequence θ = (kr ) of positive integers, such that
k0 = 0 andkr − kr−1 → ∞, is called a lacunary sequence.
We say that x = (xk ) is lacunary statistically convergent to α
if for each openset U containing α, we have
limr→∞
|{k ∈ (kr−1, kr ] : xk 6∈ U}|kr − kr−1
= 0.
The issue is: what is the relationship between the gaps of a
lacunary versionof a summability method and the gaps of the
corresponding Tauberiantheorem?
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
Proposition
Let X be a T2 space, and let θ be any lacunary sequence. Then
the followingstatements are equivalent.
Every X-valued C1-statistically convergent sequence is also
θ-lacunarystatistically convergent.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
Proposition
Let X be a T2 space, and let θ be any lacunary sequence. Then
the followingstatements are equivalent.
Every X-valued C1-statistically convergent sequence is also
θ-lacunarystatistically convergent.
lim infr (kr+1 − kr )/kr > 0.
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
Proposition
Let X be a T2 space, and let θ be any lacunary sequence. Then
the followingstatements are equivalent.
Every X-valued C1-statistically convergent sequence is also
θ-lacunarystatistically convergent.
lim infr (kr+1 − kr )/kr > 0.
Note that the second condition happens to be the same as the
gap-Tauberiancondition for the Cesàro method (C1).
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Outline Summability Methods The Setup A Bit of History Abelian
Side Tauberian Side
Lacunary vs. gap rates
Proposition
Let X be a T2 space, and let θ be any lacunary sequence. Then
the followingstatements are equivalent.
Every X-valued C1-statistically convergent sequence is also
θ-lacunarystatistically convergent.
lim infr (kr+1 − kr )/kr > 0.
Note that the second condition happens to be the same as the
gap-Tauberiancondition for the Cesàro method (C1).
This raises the issue if similar results can be constructed for
generalA-statistical convergence and their lacunary counterparts.
This is also still anopen problem.
OutlineSummability Methods
The Setup
A Bit of History
Abelian Side
Tauberian Side